Eikonal depth: an optimal control approach to statistical depths

TL;DR

This paper introduces a novel statistical depth measure based on control theory and eikonal equations, offering an interpretable, computationally feasible, and robust way to analyze high-dimensional and non-Euclidean data distributions.

Contribution

It proposes a new globally defined statistical depth using control theory, extending to non-Euclidean data and demonstrating robustness against adversarial perturbations.

Findings

Depth captures multi-modal behavior effectively

Method is robust under approximate isometric adversarial models

Illustrated with mixture models and MNIST examples

Abstract

Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.